Complex Numbers
Roots of Unity
Grade Class 11

Question:

<p>Let \(\omega\) be a primitive cube root of unity. Which are always true?</p>
\(1+\omega+\omega^2=0\)
\(\omega^3=1\)
\(\bar{\omega}=\omega^2\)
\(|\omega|=1\)

Step-by-Step Solution

Key Concept: All four are standard properties of primitive cube roots of unity. (A) minimal polynomial; (B) definition; (C) \omega = e^(2\pi i/3), conjugate = e^(-2\pi i/3) = e^(4\pi i/3) = \omega^2; (D) |e^(2\pi i/3)| = 1.
<p>All four are fundamental properties: (A) ✓, (B) ✓, (C) ✓ (\(\bar{\omega} = e^{-2\pi i/3} = e^{4\pi i/3} = \omega^2\)), (D) ✓.</p>
Correct Answer: ABCD

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